
theorem Th9:
  for L being complete Lattice for a being Element of L holds a [=
  a*' & *'a [= a
proof
  let L be complete Lattice;
  let a be Element of L;
  set X = {d where d is Element of L : a [= d & d <> a};
  for q being Element of L st q in X holds a [= q
  proof
    let q be Element of L;
    assume q in X;
    then ex q9 being Element of L st q9 = q & a [= q9 & q9 <> a;
    hence thesis;
  end;
  then
A1: a is_less_than X by LATTICE3:def 16;
  set X = {d where d is Element of L : d [= a & d <> a};
  for q being Element of L st q in X holds q [= a
  proof
    let q be Element of L;
    assume q in X;
    then ex q9 being Element of L st q9 = q & q9 [= a & q9 <> a;
    hence thesis;
  end;
  then X is_less_than a by LATTICE3:def 17;
  hence thesis by A1,LATTICE3:34,def 21;
end;
